How to use this calculator
Body 1 starts on the left and body 2 on the right, and positive velocity points right. Enter each mass and velocity, using a negative velocity for something moving left. Choose Elastic for a perfectly bouncy impact, Stick together for a perfectly inelastic one, or Choose e to enter a coefficient of restitution between 0 and 1.
Formulas
With e = 1 these reduce to the elastic-collision formulas, and kinetic energy is conserved. With e = 0 both bodies leave at the same velocity, (m₁v₁ + m₂v₂)/(m₁ + m₂), and the most kinetic energy that momentum allows is lost.
Worked example
2 kg at 3 m/s hits 1 kg coming the other way at 1 m/s (the default)
- Total momentum: 2 × 3 + 1 × (−1) = 5 kg·m/s.
- Elastic: v₁′ = (5 + 1 × 1 × (−4))/3 = 0.333 m/s and v₂′ = (5 + 2 × 1 × 4)/3 = 4.333 m/s. Kinetic energy stays at 9.5 J.
- Stick together instead: both move at 5/3 = 1.667 m/s, and 5.33 J (56% of the kinetic energy) is lost.
Typical coefficients of restitution
e describes how much of the closing speed comes back as separation speed. A superball on concrete is about 0.85 to 0.9, a basketball about 0.75 to 0.85, a baseball on a wooden bat roughly 0.5 to 0.55, steel on steel 0.6 to 0.9 depending on speed, and a car crash at highway speed close to 0, because the crumple zones absorb the energy. Measure it by dropping an object from height h and measuring the rebound height h′: e = √(h′/h).
Assumptions and limits
- Head-on (central) impact along one line, with no spin and no friction between the bodies.
- No outside forces during the brief impact, so momentum is conserved. Gravity and friction act over a much longer time and can be ignored during contact.
- e is a property of the pair of materials and the impact speed; it is not a constant for one object alone.
Frequently asked questions
Why do equal masses swap velocities?
In an elastic collision between equal masses the formulas give v₁′ = v₂ and v₂′ = v₁. That’s the Newton’s cradle effect: the moving ball stops and the struck ball carries on.
Where does the lost energy go?
Into heat, sound, vibration and permanent deformation. Momentum is always conserved; kinetic energy is conserved only when e = 1.
What is impulse?
The change in momentum of one body, equal to the average contact force times the contact time. Body 2 receives an impulse equal and opposite to body 1’s.